In this category are included any equation or set of equations where the unknowns must be integer numbers. It includes Fermat Last Theorem as a special case....
This page was last updated on October 8th, 2008
This page was last updated on October 8th, 2008
Diophantine Equations (from Number Theory)
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Rational and Integral Points on Higher-dimensional Varieties
Some of conjectures and open problems, compiled at AIM.
Developing A General 2nd Degree Diophantine Equation x^2 + p = 2^n
Methods to solve these equations.
Quadratic Diophantine Equation Solver
Dario Alpern's Java/JavaScript code that solves Diophantine equations of the form Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 in two selectable modes: "solution only" and "step by step" (or "teach") mode. There is also a link to his
Diophantine Equations
Dave Rusin's guide to Diophantine equations.
The Erdos-Strauss Conjecture
The conjecture states that for any integer n > 1 there are integers a, b, and c with 4/n = 1/a + 1/b + 1/c, a > 0, b > 0, c > 0. The page establishes that the conjecture is true for all integers n, 1 < n <= 10^14. Tables and software by
Egyptian Fractions
Lots of information about Egyptian fractions collected by David Eppstein.
Thue Equations
Definition of the problem and a list of special cases that have been solved, by Clemens Heuberger.
Linear Diophantine Equations
A web tool for solving Diophantine equations of the form ax + by = c.
Hilbert's Tenth Problem
Given a Diophantine equation with any number of unknowns and with rational integer coefficients: devise a process, which could determine by a finite number of operations whether the equation is solvable in rational integers.
Diophantus Quadraticus
On-line Pell Equation solver by Michael Zuker.
On the Psixyology of Diophantine Equations
PhD thesis, Pieter Moree, Leiden, 1993.
Pythagorean Triples in JAVA
A JavaScript applet which reads a and gives integer solutions of a^2+b^2 = c^2.
1, 3, 8, 120, ...
Sets of numbers such that the product of any two is one less than a square. Diophantus found the rational set 1/16, 33/16, 17/4, 105/16; Fermat the integer set 1, 3, 8, 120.
Fermat's Method of Infinite Descent
Notes by Jamie Bailey and Brian Oberg. Illustrates the method on FLT with exponent 4.
MAGMA program
MAGMA code to solve Diophantine equations of the form F(x)=G(y), for which Runge's condition is satisfied. Created by Szabolcs Tengely.
Diophantine Geometry in Characteristic p
A survey by José Felipe Voloch.
Diagonal Quartic Surfaces
Articles, computations and software in Magma and GP by Martin Bright.
Pell's Equation
Record solutions.
Solving General Pell Equations
John Robertson's treatise on how to solve Diophantine equations of the form x^2 - dy^2 = N.
Pythagorean Triplets
A Javascript calculator for pythagorean triplets.
Bibliography on Hilbert's Tenth Problem
Searchable, ~400 items.
Rational Triangles
Triangles in the Euclidean plane such that all three sides are rational. With tables of Heronian and Pythagorean triples.
Diophantine m-tuples
Sets with the property that the product of any two distinct elements is one less than a square. Notes and bibliography by Andrej Dujella.
Some of conjectures and open problems, compiled at AIM.
Methods to solve these equations.
Dario Alpern's Java/JavaScript code that solves Diophantine equations of the form Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 in two selectable modes: "solution only" and "step by step" (or "teach") mode. There is also a link to his
Dave Rusin's guide to Diophantine equations.
The conjecture states that for any integer n > 1 there are integers a, b, and c with 4/n = 1/a + 1/b + 1/c, a > 0, b > 0, c > 0. The page establishes that the conjecture is true for all integers n, 1 < n <= 10^14. Tables and software by
Lots of information about Egyptian fractions collected by David Eppstein.
Definition of the problem and a list of special cases that have been solved, by Clemens Heuberger.
A web tool for solving Diophantine equations of the form ax + by = c.
Given a Diophantine equation with any number of unknowns and with rational integer coefficients: devise a process, which could determine by a finite number of operations whether the equation is solvable in rational integers.
On-line Pell Equation solver by Michael Zuker.
PhD thesis, Pieter Moree, Leiden, 1993.
A JavaScript applet which reads a and gives integer solutions of a^2+b^2 = c^2.
Sets of numbers such that the product of any two is one less than a square. Diophantus found the rational set 1/16, 33/16, 17/4, 105/16; Fermat the integer set 1, 3, 8, 120.
Notes by Jamie Bailey and Brian Oberg. Illustrates the method on FLT with exponent 4.
MAGMA code to solve Diophantine equations of the form F(x)=G(y), for which Runge's condition is satisfied. Created by Szabolcs Tengely.
A survey by José Felipe Voloch.
Articles, computations and software in Magma and GP by Martin Bright.
Record solutions.
John Robertson's treatise on how to solve Diophantine equations of the form x^2 - dy^2 = N.
A Javascript calculator for pythagorean triplets.
Searchable, ~400 items.
Triangles in the Euclidean plane such that all three sides are rational. With tables of Heronian and Pythagorean triples.
Sets with the property that the product of any two distinct elements is one less than a square. Notes and bibliography by Andrej Dujella.
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